Soft computing approaches for forecasting reference evapotranspiration

Highlights

• The GP, SVM-FFA, ANN and SVM–Wavelet modeling of ET₀ was reported.

• SVM–Wavelet had the smallest RMSE of 0.233 mm day⁻¹ in testing phase.

• The ANN model had the largest RMSE of 0.450 mm day⁻¹.

• SVM–Wavelet model was found to perform better than the GP, SVM-FFA and ANN models.

Abstract

Accurate estimation of reference evapotranspiration (ET₀) is needed for planning and managing water resources and agricultural production. The FAO-56 Penman–Monteith equation is used to determinate ET₀ based on the data collected during the period 1980–2010 in Serbia. In order to forecast ET₀, four soft computing methods were analyzed: genetic programming (GP), support vector machine-firefly algorithm (SVM-FFA), artificial neural network (ANN), and support vector machine–wavelet (SVM–Wavelet). The reliability of these computational models was analyzed based on simulation results and using five statistical tests including Pearson correlation coefficient, coefficient of determination, root-mean-square error, absolute percentage error, and mean absolute error. The end-point result indicates that SVM–Wavelet is the best methodology for ET₀ prediction, whereas SVM–Wavelet and SVM-FFA models have higher correlation coefficient as compared to ANN and GP computational methods.

Introduction

Evapotranspiration (ET) is considered as one of the most vital elements in hydrologic cycle and has an important role in the field of water resource management. It can be measured directly or estimated by calculating reference evapotranspiration (ET₀). There have been a number of equations used to estimate ET₀. However, the FAO-56 Penman–Monteith (FAO-56 PM) equation (Allen et al., 1998) has been confirmed to be superior in comparison with other techniques that are often used as the reference equation (Gavilan et al., 2007, Lopez-Urrea et al., 2006, Tabari et al., 2013, Ventura et al., 1999). Pereira et al. (2015) discuss the past and future of the FAO-56 PM as well as the accuracy and consistency of operational computation of ET₀ for agricultural uses.

Soft computing methods can be used as alternative techniques because they offer benefits such as no required knowledge of internal system variables, simpler solutions for multi-variable problems and factual calculation (Chaturvedi, 2008, Huang et al., 2010, Zadeh, 1992). Soft computing is an innovative approach in constructing computationally intelligent systems. According to Zadeh (1992), soft computing is an emerging method towards computing which parallels the significant ability of the human intelligence to comprehend in an environment of imprecision and uncertainty.

Nowadays, soft computing methods have shown to be superior and reliable in estimating and forecasting ET₀. Kumar et al. (2011) discussed the artificial neural network (ANN) in creating a model of the evapotranspiration process and selecting training algorithm in estimating and forecasting ET₀. The ANN models for ET₀ are improvised via various learning algorithms like radial basis function (RBF) that was applied by Kim and Kim (2008) and Kisi (2006) during the implementation of the generalized regression neural networks (GRNN), back-propagation (Kumar et al., 2002, Kisi, 2006), Levenberg–Marquardt algorithm (LMA) also known as the damped least-squares (DLS) method (Khoob, 2008, Kisi, 2007, Zanetti et al., 2007) and conjugate gradient decent (Landeras et al., 2008). During the development of the ANN models, different categories of ET₀ estimation methods have been considered (Trajkovic et al., 2000, Kisi, 2007, Khoob, 2007, Zanetti et al., 2007, Kumar et al., 2008, Kumar et al., 2009, Landeras et al., 2008, Trajkovic, 2010, Citakoglu et al., 2014).

Genetic programming (GP) is used alongside with ANNs and other soft computing to create simulated hydrological processes like ET without knowing the direct relation between separate components involved (Aghajanloo et al., 2013, Kim and Kim, 2008, Guven et al., 2008, Irmak and Kamble, 2009, Kisi and Guven, 2010, Kisi and Cengiz, 2013, Parasuraman et al., 2007). Recently, explicit mathematical formulation of ET₀ can be done by using gene-expression programming (GEP) (Traore and Guven, 2012, Traore and Guven, 2013, Shiri et al., 2012, Shiri et al., 2014a).

Support vector machine (SVM) (Vapnik, 1995, Vapnik, 1998) is one of the novel soft learning algorithms that has been recently realized for a wide range of applications in the field of soft computing, hydrology and environmental studies (Lee and Verri, 2003, Lu and Wang, 2005, Asefa et al., 2006, Ji and Sun, 2013, Sun, 2013). It is used in pattern recognition, classification, forecasting and regression analysis which has proven to have superior performance over previously developed methodologies; such as neural network and other conventional statistical models (Vapnik et al., 1997, Joachims, 1998, Collobert and Bengio, 2000, Huang et al., 2002, Mukkamala et al., 2002, Sung and Mukkamala, 2003). SVM has gained importance in forecasting problems of ET₀ (Kisi and Cimen, 2009, Kisi, 2013, Shiri et al., 2014b).

SVM was created in accordance with the standard statistical learning processes and structural risk minimization. This reduces the upper bound generalization error rather than the local training error. The latter is the common route used in conventional machine learning methods (Vapnik, 1998). This shows the superiority of SVM compared with other soft learning algorithms.

The following criteria for SVM modeling can be defined: first, the inclusion of the novel solution because of the convex nature of the optimal problem and second, utilization of high dimensional-space sets of kernel functions that discreetly hold non-linear transformation. In essence, there are no presumptions in functional transformation that makes data linearly unique. A linearly separable problem is one in which a decision hyper plane can be found in the input space separating the input patterns with desired output. Application of kernel functions enables mapping of input data points that are not linearly separable into higher dimensional space, which makes them linearly separable.

In the past decades, wavelet transform (WT) has been applied in the field of ET₀ modeling. WT captures both frequency and location information (location in time) (DeVore et al., 1992) and has some desirable properties compared to the Fourier transform (Strang, 1993). The transform is based on a wavelet matrix, which can be computed more quickly than the analogous Fourier matrix (Strang, 1993). Kisi (2011) combined linear regression model and discrete wavelet transform (DWT) to model ET₀ and realized that the wavelet regression (WR) approach performs better than conventional empirical models in daily ET₀ modeling. Cobaner (2013) applied WR method in estimating ET₀ based on Class A pan evaporation, while Partal, 2009, Falamarzi et al., 2014 used a form of wavelet neural network to estimate ET₀.

Our study uses forecasting model to anticipate ET₀ using SVM and firefly algorithm (FFA) (Yang, 2010a, Yang, 2010b). FFA is a nature-inspired metaheuristic optimization algorithm that works by the flashing behavior of fireflies (Yang, 2009). The primary purpose for a firefly’s flash is to act as a signal system to attract other fireflies. The FFA is used to define the optimal SVM parameters so SVM and FFA are merged in this study to enable more superior performance of SVM model (SVM-FFA). SVM is also merged with WT algorithm (SVM–Wavelet) to capture frequency and location information from the ET₀ data. The objectives of the current study are to construct, develop and evaluate the results of SVM–Wavelet, SVM-FFA, GP and ANN for ET₀ prediction.